×

Geodesics on hypersurfaces in Minkowski space: singularities of signature change. (English. Russian original) Zbl 1254.53071

Russ. Math. Surv. 66, No. 6, 1201-1203 (2011); translation from Usp. Mat. Nauk 66, No. 6, 193-194 (2011).
From the text: We announce new results on geodesics in degenerate (singular) metrics (see the preceding papers [A. O. Remizov, Sb. Math. 200, No. 3, 385–403 (2009); translation from Mat. Sb. 200, No. 3, 75–94 (2009; Zbl 1169.53318); Proc. Steklov Inst. Math. 268, 248–257 (2010); translation from Trudy Mat. Inst. Steklova 268, 258–267 (2010; Zbl 1200.58026); Russ. Math. Surv. 65, No. 1, 180–182 (2010); translation from Usp. Mat. Nauk. 65, No. 1, 187–188 (2010; Zbl 1194.53065)], and [D. Genin, B. Khesin and S. Tabachnikov, Enseign. Math. (2) 53, No. 3–4, 307–331 (2007; Zbl 1144.37013) and B. Khesin and S. Tabachnikov, Adv. Math. 221, No. 4, 1364–1396 (2009; Zbl 1173.37037)]).
We consider geodesics on a hypersurface in Minkowski space, that is, four-dimensional pseudo-Euclidean space with signature \(+++-\). The metric in the ambient space induces on the hypersurface a metric \(ds^2\) which is Riemannian (signature \(+++\)) in one open domain and pseudo-Riemannian (signature \(++-\)) in another open domain. These domains are separated by a two-dimensional surface on which the metric \(ds^2\) is degenerate (its determinant \(\Delta\) vanishes), and the points of this surface are called signature changing points. The equation of the geodesic flow generated by \(ds^2\) has singularities at signature changing points that lead to a curious phenomenon: geodesics cannot pass through such a point in arbitrary tangential directions, but only in certain directions said to be admissible. The number of geodesics passing through a given signature changing point in different admissible directions can also vary. The present note is devoted to the study of this phenomenon.

MSC:

53C22 Geodesics in global differential geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
34A26 Geometric methods in ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI